/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 70 Solve the inequality and graph i... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 70

Solve the inequality and graph its solution. \(-3>2 x-5\)

Short Answer

Expert verified
The solution to the inequality \(-3 > 2x - 5\) is \(x < 1\). This is represented on a number line with an open circle at 1 and an arrow pointing left, indicating values less than 1.

Step by step solution

01

Addition/subtraction

First, the constant from the right hand side must be moved to the left hand side of the inequality. This can be done by adding 5 to both sides. So, the inequality -3 > 2x - 5 becomes (-3+5) > 2x, simplifying to 2 > 2x.
02

Division

Divide each side by 2 (the coefficient of x) to isolate x: \(2/2 > 2x/2\). This results in 1 > x.
03

Swap sides

Normally, we write this with 'x' on the left hand side: x < 1.
04

Graph the solution

To graph the solution, draw a number line. The solution x < 1 indicates that x can be any number less than 1. On the number line, a open circle (because x cannot equal to 1) is placed at 1, and an arrow is drawn pointing towards the left to denote 'less than'.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality Solving
Solving inequalities is a fundamental algebraic skill that allows you to find the set of solutions that satisfy certain relationships. In inequalities, unlike equations, you deal with expressions where one side is not strictly equal to the other. Instead, symbols like \(<, >, \leq, \geq\) are used to denote the nature of the relationship between the expressions.
To solve inequalities, follow these steps:
  • Move terms around to isolate the variable of interest. This often involves using addition or subtraction.
  • Divide or multiply both sides by a coefficient to fully isolate the variable. Be cautious: multiplying or dividing by a negative number reverses the inequality sign.
  • Simplify your inequality expression to reach the simplest form.

In applied problems, always recheck the direction of the inequality sign to make sure it follows logical steps.
Graphing Inequalities
Graphing inequalities is a powerful way to visually interpret solutions. Once you find the inequality solution, such as \(x < 1\), translating it into a graph is helpful for understanding.
Start by visualizing the solution on a coordinate system. On a one-dimensional number line, positions to the left of the evaluated point represent numbers less than that point, whereas positions to the right represent greater values.
  • Identify the boundary number (in this case, 1 for \(x < 1\)).
  • Determine whether the boundary is included by checking if the inequality is strict (\(< , >\) for not included) or inclusive (\(\leq , \geq\) for included).
  • Use symbols like open circles or closed dots on the boundary to show whether it's part of the set.

Graphs provide an intuitive way to check solutions and ensure the inequality is accurately depicted.
Number Line Representation
Number lines are a simple yet effective way to visually represent inequalities. They help in understanding the scope and scale of numbers involved in inequality problems.
When representing an inequality on a number line:
  • Start by drawing a simple horizontal line. This will serve as your number line.
  • Mark relevant points on the line, such as integers or other numbers that are significant to your problem.
  • If the inequality uses a boundary number that's not part of the solution set (like \(x < 1\)), use an open circle above 1.
  • Draw a shaded line or arrow from the open circle extending towards the side of the line that includes the solution set. For \(x < 1\), this will be to the left.

Through this process, number lines clearly show which parts of the number spectrum satisfy the inequality. This visual representation is especially useful when communicating solutions in math.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Factor the expression. $$48 y^{2}-72 x y+27 x^{2}$$

Find the product. $$(4 t-1)^{2}$$

Factor the expression. Tell which special product factoring pattern you used. $$z^{2}-25$$

An object is propelled from the ground with an initial upward velocity of 224 feet per second. Will the object reach a height of 784 feet? If it does, how long will it take the object to reach that height? Solve by factoring.

Which one of the following is a correct factorization of the expression \(-16 x^{2}+36 x+52 ?\). A) \((-16 x+26)(x-2)\) (B) \((-4 x-13)(4 x+4)\) (C) \(-1(4 x-13)(4 x+4)\) (D) \(-1(4 x-4)(4 x+13)\)
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.